I am trying to understand the relation between two different presentations of the Simplex Algorithm:
First Presentation:
Let $x,c \in \mathbb{R}^{n}$, $A \in \mathbb{R}^{n \times m}$, $b \in \mathbb{R}^{m}$, and let us assume we have a column basis B, such that $A=(A_B A_N)$, where $N = \{1,\ldots,m\} \setminus B$. We can write $x=(x_{B}x_{N})$ accordingly.
Then we can define for some $j \in N$ a vector $d \in \mathbb{R}^{n}$ with $d_{B} = -B^{-1}A_{j}$, $d_{j} = 1$ and $d_{i} = 0 \, \forall i \in N \setminus\{j\}$.
For $\theta \in \mathbb{R}_{> 0 }$, we define $y = x + \theta d$. We have $c^{T}y -c^{T}x = \theta [c_{j} - c_{B}^{T}B^{-1}A_{j}] := \theta \overline{c}_{j}$. Now if $\overline{c}_{j} <0 $, we have $c^{T}y < c^{T}x$. We can show that $Ay = b$.
If $d_{i} <0 $ for some $i$, then we can choose $\theta \leq -x_{i}/d_{i}$, which ensures that $y_{i} >= 0$. Setting $\theta = \min_{i : d_{i} < 0 } -x_{i}/d_{i}$ thus ensures that $y \geq 0$ which makes y a valid solution that improves the objective value.
Second Presentation:
Now the other presentation I see is with the Simplex tableau. Here (for maximization) we search for an index j such that $c_{j}>0$ is maximal. Then we search for the entry i such that $l_{i}:=b_{i}/A_{i,j}$ is minimal and positive ($\hat{i} = \arg \min_{i: A_{i,j}> 0} l_{i} )$, which gives an optimal row $\hat{i}$. Now we divide the row $A_{\hat{i},:}$ by $A_{\hat{i},j}$ and apply gauss transformations to transform column j to a basis column, i.e. such that it corresponds to the canonical basis vector $e_{\hat{i}}$.
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The calculations look similar but they are not the same, and I don't exactly see how the two presentations relate. Can someone provide more insights?